New approach to machine learning optimization using distribution space.
problem Optimization challenges in machine learning with non-convex constraints.
method Relaxation to convex optimization in distribution space, numerical algorithm based on mixture distributions.
result Approximate optimization in distribution space is consistent and effective.
Concrete distribution properties examined on simplex.
problem Properties of Concrete distribution on simplex.
method Reflection and location-scale transformation of uniform distribution; explicit parameterization to Poincaré half-space.
result Fisher information and information metric are hyperbolic space; Fisher-Rao geodesic distance computed.
The paper introduces exponential-wrapped distributions on symmetric spaces for better data modeling.
problem Challenges in statistical modeling due to curvature of data spaces.
method Construction and use of exponential-wrapped distributions on affine locally symmetric spaces.
result Exponential-wrapped distributions on symmetric spaces have useful properties for practical use.
Smooth distributions on subcartesian spaces can be globally finitely generated.
problem Understanding smooth distributions on subcartesian spaces.
method Embedding in Euclidean space, Whitney Embedding Theorem, and distribution theory.
result Smooth generalized distributions and subbundles on connected subcartesian spaces are globally finitely generated.
We study diffeologies on locally convex spaces and their application to smooth multiplication of distributions.
problem Constructing smooth multiplication of distributions on locally convex spaces.
method Using diffeological colimits and wavefront-set criterion.
result Proving smooth multiplication of microlocally multipliable distributions.
Study on diffeologies on locally convex spaces and smooth multiplication of distributions.
problem Geometric characterization and smoothness of distribution multiplication.
method Investigation of canonical and c∞-diffeologies on locally convex spaces, proving geometric characterizations, and comparing diffeologies. result Established a framework for nonlinear distribution theory beyond manifolds, realizing microlocally multipliable distributions as a diffeological colimit.
We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves, and show that codimension one distr…
Hyperbolic space is a geometry that is known to be well-suited for representation learning of data with an underlying hierarchical structure. In this paper, we present a novel hyperbolic distribution called \textit{pseudo-hyperbolic Gaussian}, a Gaussian-like distribution on hyperbolic space whose density can be evalua…
This paper studies the Fisher-Rao geometry on the parameter space of beta distributions. We derive the geodesic equations and the sectional curvature, and prove that it is negative. This leads to uniqueness for the Riemannian centroid in that space. We use this Riemannian structure to study canonical moments, an intrin…
A Riemannian symmetric space is a Riemannian manifold in which it is possible to reflect all geodesics through a point by an isometry of the space. On such spaces, we introduce the notion of a distributional lattice, generalizing the notion of lattice. Distributional lattices exist in any Riemannian symmetric space: th…
Study finds homogeneous spaces with geodesic orbits but no integrable distributions.
problem Characterizing homogeneous spaces with specific geometric properties.
method Examined Lie groups with compact stabilizers and classified spaces based on geodesic orbits and integrable distributions.
result Identified homogeneous spaces with geodesic orbits but lacking integrable invariant distributions.
Variational Auto-Encoders enforce their learned intermediate latent-space data distribution to be a simple distribution, such as an isotropic Gaussian. However, this causes the posterior collapse problem and loses manifold structure which can be important for datasets such as facial images. A GAN can transform a simple…
AI learns to classify and represent univariate distributions in a 2D latent space.
problem Classifying and representing univariate empirical distributions.
method Unsupervised beta variational autoencoder (beta-VAE) to separate and represent distributions in a 2D latent space.
result The latent space representation separates distributions of different shapes while overlapping similar ones.
In this paper the result of real hypersurfaces in non-flat complex space forms, whose structure vector field ξ belongs to the κ-nullity distribution is extended in case of three dimensional real hypersurfaces in non-flat complex space forms. Furthermore, generalization of notion (κ,μ)-nullity distribution defin…
New method for reducing dimensions of distributional data.
problem Nonlinear sufficient dimension reduction for distribution-on-distribution regression.
method Building universal kernels on metric spaces to characterize conditional independence.
result Method outperforms competing methods in synthetic and real data applications.
New probabilistic invariants bound classical topological complexity and category.
problem Bounding classical topological complexity and category.
method Developed probabilistic variants of one-category and diagonal topological complexity.
result Identified new invariants with distributional category and complexity on Eilenberg-Mac Lane spaces.
The paper studies distributions on surfaces and their connection to twistor spaces.
problem Understanding distributions invariant under geodesic flows on surfaces.
method Analyzes transport equations on unit tangent bundles and connects to twistor spaces.
result Holomorphic distributions form a unital algebra and are bijectively related to functions on twistor space.
This work improves policy optimization by maximizing entropy of state distribution, leading to better exploration.
problem Lack of exploration in state space when maximizing policy entropy.
method Proposes maximizing the entropy of a lower bound approximation to the state weighting distribution, based on latent space representation.
result Entropy regularization based on marginal state distribution achieves superior state space coverage and better performance in various domains.
LSDM uses unpaired data to match latent space distributions for generative modeling.
problem Generating high-quality images with limited paired data.
method Two-stage approach: latent space learning from paired and unpaired data, followed by joint distribution matching.
result LSDM enhances geometric fidelity in generated outputs and provides theoretical insights into LDMs.
A new method embeds distributions in a common space for optimal transport comparison.
problem Comparing distributions in different metric spaces.
method Sub-embedding robust Wasserstein (SERW) distance.
result SERW mimics GW distance properties and provides a cost relation.
Generative AutoEncoders require a chosen probability distribution in latent space, usually multivariate Gaussian. The original Variational AutoEncoder (VAE) uses randomness in encoder - causing problematic distortion, and overlaps in latent space for distinct inputs. It turned out unnecessary: we can instead use determ…
The study classifies and analyzes two-dimensional holomorphic distributions on a four-dimensional projective space.
problem Classifying and analyzing two-dimensional holomorphic distributions on P4. method Classification and investigation of distributions with specific properties, including tangent and conormal sheaves.
result The sheaves of distributions are split and the moduli spaces are irreducible quasi-projective varieties.
The present paper unifies some aspects concerning the vertical Liouville distributions on the tangent (cotangent) bundle of a Finsler (Cartan) space in the context of generalized geometry. More exactly, we consider the big-tangent manifold TM associated to a Finsler space (M,F) and of its L-du…
Study risk bounds for distributed ERM with general loss functions and hypothesis spaces.
problem Limited theoretical analysis for distributed ERM with general loss functions and hypothesis spaces.
method Derive tight risk bounds under assumptions on hypothesis space and loss function.
result Developed more general risk bound for distributed ERM without strong convexity restriction.
New framework models graph signals as distribution-valued signals in Wasserstein space.
problem Limitations of classical vector-based GSP, including synchronous observations and uncertainty.
method Introduces graph distribution-valued signals (GDSs) in the Wasserstein space.
result GDSs naturally encode uncertainty and stochasticity, generalizing traditional graph signals.
Paper proposes a framework to detect distribution shifts using embedding space geometry.
problem Detecting distribution shifts in candidate datasets to improve model generalizability.
method Non-parametric framework using embedding space geometry for two tests: robustness boundary and in-distribution/out-of-distribution classification.
result Both tests successfully detect distribution shifts in various scenarios for both synthetic and real-world datasets.
Generative Distribution Embeddings learn multiscale representations of distributions.
problem Learning representations of entire distributions for multiscale reasoning.
method Introducing GDE framework that lifts autoencoders to the space of distributions, using conditional generative models and distributional invariance.
result GDEs learn predictive sufficient statistics embedded in Wasserstein space, recovering distances and trajectories for Gaussian and Gaussian mixture distributions.
A nonparametric approach for policy learning for POMDPs is proposed. The approach represents distributions over the states, observations, and actions as embeddings in feature spaces, which are reproducing kernel Hilbert spaces. Distributions over states given the observations are obtained by applying the kernel Bayes' …
SDPG algorithm improves sample efficiency and reward in DRL for continuous action spaces.
problem Improving sample efficiency and reward in distributional reinforcement learning for continuous action spaces.
method SDPG algorithm models return distribution using samples via reparameterization technique.
result SDPG shows better sample efficiency and higher reward in OpenAI Gym environments.
Diffusion models achieve nearly optimal distribution estimation in various spaces.
problem Theoretical limitations of diffusion modeling for distribution estimation.
method Analysis of approximation and generalization abilities of diffusion models in Besov spaces.
result Diffusion models achieve nearly minimax optimal estimation rates in total variation and Wasserstein distances.
New theory of distributions on spaces with singular submanifolds.
problem Defining distributions on spaces with singular submanifolds.
method Construction of thick distributions, operations, and special distributions.
result Clarified connection between thick and classical distributions.
A new method estimates multi-dimensional value distributions using Hilbert space embeddings.
problem Estimating value distributions in complex, multi-dimensional reinforcement learning settings.
method Hilbert space mappings and kernel mean embeddings to estimate the kernel mean embedding of multi-dimensional value distributions.
result Uniform convergence guarantees and robust off-policy evaluation demonstrated in simulations.
A pivotal problem in Bayesian nonparametrics is the construction of prior distributions on the space M(V) of probability measures on a given domain V. In principle, such distributions on the infinite-dimensional space M(V) can be constructed from their finite-dimensional marginals---the most prominent example being the…
Proposes PER loss to regularize neural network activations to normal distribution.
problem Improving neural network generalization and training speed.
method Regularizes activations to standard normal distribution via projected error function and Wasserstein distance.
result Minimizes Wasserstein distance between activation distribution and standard normal.
In this paper we investigate codimension one Fano distributions on Fano manifolds with Picard number one. We classify Fano distributions of maximal index on complete intersections in weighted projective spaces, Fano contact manifolds, Grassmannians of lines and their linear sections, and describe their moduli spaces. A…
Generative models such as Variational Auto Encoders (VAEs) and Generative Adversarial Networks (GANs) are typically trained for a fixed prior distribution in the latent space, such as uniform or Gaussian. After a trained model is obtained, one can sample the Generator in various forms for exploration and understanding,…
We show a duality which arises from distributions of Cartan type, having growth (2, 3, 5), from the view point of geometric control theory. In fact we consider the space of singular (or abnormal) paths on a given five dimensional space endowed with a Cartan distribution, which form another five dimensional space with a…
This work proposes a new method to match distributions across different spaces using cycle-consistent maps.
problem Matching distributions across different spaces with consistent bidirectional maps.
method A novel unbalanced Monge optimal transport formulation for matching distributions on different spaces, employing cycle-consistent maps.
result The proposed discrepancy captures the cycle-consistent GAN framework and provides theoretical support.
UAPCA projects uncertain data to low dimensions using GMMs.
problem Uncertain multidimensional data not well described by normal distributions.
method Model data with Gaussian mixture models, derive UAPCA projection from general formulation.
result Low-dimensional projections better represent multidimensional distributions.
Study of ants' movement rules on a 6D space, revealing distribution structures and singular trajectories.
problem Understanding the movement patterns of ants in a 6D space.
method Analyzing mechanical system rules to derive distribution structures and singular trajectories.
result Distributions and singular trajectories of ants' movement rules in a 6D space.
The paper analyzes variational autoencoders for state space models with risk bounds.
problem Analyzing the risk associated with variational autoencoders for state space models.
method Backward factorization of variational distributions to analyze excess risk, providing oracle inequalities and upper bounds.
result Explicit upper bounds on variational estimation error for state space models under strong mixing assumptions.
Study on geometry of Dirichlet distributions using Fisher-Rao metric.
problem Understanding the geometry of Dirichlet distributions.
method Analysis of Fisher-Rao metric on Dirichlet distribution parameter space.
result Geodesic completeness and negative sectional curvature of the space.
New bounds show triangulated surfaces are evenly distributed in moduli space.
problem Distribution of triangulated surfaces in moduli space as genus increases.
method Proved upper and lower bounds for the number of triangulated surfaces in Teichmüller balls.
result Number of triangulated surfaces in a Teichmüller unit ball is at most exponential in the number of triangles, independent of genus.
LOT embeds distributions for linear separability and classification.
problem Distribution discrimination in various scientific fields.
method Linear Optimal Transport (LOT) embedding into L2 space. result LOT embeds distributions into linearly separable spaces for certain transformations and perturbations.
FTIP uses normalizing flows to improve posterior inference in function space.
problem Challenges in posterior inference with implicit-process priors.
method FTIP uses normalizing flows to define a richer variational distribution over combination weights.
result FTIP captures asymmetric and multimodal posterior structure better than Gaussian coefficient approximations.
The method approximates stationary distributions of Markov models by truncating irrelevant states.
problem Computing the stationary distribution of complex Markov models is computationally challenging.
method A state-space lumping scheme that aggregates states in a grid structure, iteratively refining the state-space.
result The method provides a well-justified finite-state projection tailored to the stationary behavior of Markov models.
Paper establishes DRL for high-dimensional rewards.
problem Intractable reinforcement learning with high-dimensional rewards.
method Theoretical foundations and a novel DRL algorithm.
result Bellman operator contraction in high-dimensional spaces.
Sharp bounds for max-sliced Wasserstein distances derived for empirical distributions.
problem Estimating the expected max-sliced Wasserstein distance between a probability measure and its empirical distribution.
method Banach space version and operator norm approach for upper bounds.
result Upper bounds for max-sliced Wasserstein distances are essentially matching and sharp up to a log factor.